arXiv subjects
Publications and source records attributed to Shu Kawaguchi.
There exists a semipositive adelic line bundle on a projective variety over $\mathbb{Q}$ whose absolute minimum does not attain the minimum by algebraic points. The construction is based on arithmetic properties of a family of dynamical systems over $\mathbb{P}^1$.
Given a divisor on a tropical curve, we associate to each point of the curve a Weierstrass gap sequence. We investigate structural properties of these gap sequences and explore their relationship with the Weierstrass gap sequences of line bundles on algebraic curves via the tropicalization process.
We show that there are uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension $g\geqslant1$, generalizing a result of Hultberg in dimension 1.
We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over nonarchimdean valued fields. In the course of this work, we introduce the notion of a rational polyhedral orbifold and we provide faithful embeddings of tropical Kummer surfaces as such. Further, we show faithful tropicalizations of the canonical skeletons of certain Kummer surfaces over nonarchimdean valued fields. Under a suitable assumption on the base field, the canonical skeletons coincide with the Kontsevich--Soibelman skeletons.
Let $A$ be an abelian variety over an algebraically closed field $k$ that is complete with respect to a nontrivial nonarchimedean absolute value. Let $A^{\mathrm{an}}$ denote the analytification of $A$ in the sense of Berkovich, and let $Σ$ be the canonical skeleton of $A^{\mathrm{an}}$. In this paper, we obtain a faithful tropicalization of $Σ$ by nonarchimedean theta functions, giving a tropical version of the classical theorem of Lefschetz on abelian varieties. Key ingredients of the proof are (1) faithful embeddings of tropical abelian varieties by tropical theta functions and (2) lifting of tropical theta functions to nonarchimedean theta functions, and they will be of independent interest. For (1), we use some arguments similar to the case of complex abelian varieties as well as Voronoi cells of lattices. For (2), we use Fourier expansions of nonarchimedean theta functions over the Raynaud extensions of abelian varieties.
Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton. Our main result offers a purely polyhedral characterization of this semifield: we show that a tropical function is in the image of the tropicalization map if and only if it takes the same slope near infinity along parallel half-lines of the skeleton. This extends a result of Baker and Rabinoff in dimension one to arbitrary dimensions. We use this characterization to establish that this semifield is finitely generated over the semifield of tropical rational numbers, providing a new proof of a recent result by Ducros, Hrushovski, Loeser and Ye in the discrete valued field case. As a second application, we present a new proof of the faithful tropicalization theorem by Gubler, Rabinoff and Werner in the discrete valuation field case. The proof is constructive and provides explicit coordinate functions for the embedding of the skeleton, extending the existing results in dimension one to arbitrary dimensions.
We give a formula that relates the difference of the j-invariants with the Borcherds Phi-function, an automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor.
In this paper we study arithmetic properties of a one-parameter family ${\mathbf H}$ of Hénon maps over the affine line. Given a family of initial points ${\mathbf P}$ satisfying a natural condition, we show the height function $h_{\mathbf P}$ associated to ${\mathbf H}$ and ${\mathbf P}$ is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We then show various local properties of $h_{\mathbf P}$. Next we consider the set $Σ({\mathbf P})$ consisting of periodic parameter values, and study when $Σ({\mathbf P})$ is an infinite set or not. We also study unlikely intersections of periodic parameter values.
For a connected smooth projective curve $X$ of genus $g$, global sections of any line bundle $L$ with $°(L) \geq 2g+ 1$ give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a homeomorphism onto its image preserving integral structures (called a faithful tropicalization). Let $K$ be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that $X$ is defined over $K$ and has genus $g \geq 2$ and that $Γ$ is a skeleton (that is allowed to have ends) of the analytification $X^{\mathrm{an}}$ of $X$ in the sense of Berkovich. We show that if $°(L) \geq 3g-1$, then global sections of $L$ give a faithful tropicalization of $Γ$ into tropical projective space. As an application, when $Y$ is a suitable affine curve, we describe the analytification $Y^{\mathrm{an}}$ as the limit of tropicalizations of an effectively bounded degree.
Let $R$ be a complete discrete valuation ring of equi-characteristic zero with fractional field $K$. Let $X$ be a connected, smooth projective variety of dimension $d$ over $K$, and let $L$ be an ample line bundle over $X$. We assume that there exist a regular strictly semistable model $\mathscr{X}$ of $X$ over $R$ and a relatively ample line bundle $\mathscr{L}$ over $\mathscr{X}$ with $\mathscr{L}|_{X} \cong L$. Let $S(\mathscr{X})$ be the skeleton associated to $\mathscr{X}$ in the Berkovich analytification $X^{\mathrm{an}}$ of $X$. In this article, we study when $S(\mathscr{X})$ is faithfully tropicalized into tropical projective space by the adjoint linear system $|L^{\otimes m} \otimes ω_X|$. Roughly speaking, our results show that, if $m$ is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S(\mathscr{X})$.
Let $\bar{G} = (G, ω)$ be a vertex-weighted graph, and $δ$ a divisor class on $G$. Let $r_{\bar{G}}(δ)$ denote the combinatorial rank of $δ$. Caporaso has introduced the algebraic rank $r_{\bar{G}}^{\operatorname{alg}}(δ)$ of $δ$, by using nodal curves with dual graph $\bar{G}$. In this paper, when $\bar{G}$ is hyperelliptic or of genus $3$, we show that $r_{\bar{G}}^{\operatorname{alg}}(δ) \geq r_{\bar{G}}(δ)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a non-hyperelliptic curve of genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.
Let $(G, ω)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, ω)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with reduction graph $(G, ω)$ such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph $(G, ω)$ in general, how the existence of such $X$ relates the Riemann--Roch formulae for $X$ and $(G, ω)$, and also how the existence of such $X$ is related to a conjecture of Caporaso.
Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n h_X(P), where the implied constant depends only on X, h_X, f, and e. As applications, we prove a fundamental inequality a_f(P) \le d_f for the upper arithmetic degree and we construct canonical heights for (nef) divisors. We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and we describe some cases for which we can prove our conjecture.
The Borcherds Phi-function is the automorphic form on the moduli space of Enriques surfaces characterizing the discriminant locus. In this paper, we give an algebro-geometric construction of the Borcherds Phi-function.
The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m'th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of P^{2d+1} over Spec(Z), and that {R_{d,m,0} : m \ge 0} is a commuting family of maps.
Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. Further, if X is an abelian variety defined over a number field and D is a nonzero nef divisor, we characterize points satisfying h_D(x)=0, and we use this characterization to prove that if the f-orbit of x is Zariski dense in X, then a_f(x) is equal to the dynamical degree of f.
Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X whose forward orbits are well-defined, there is an analogous (upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an ample Weil height on X. In an earlier paper, we proved the fundamental inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit of P is Zariski dense. In this paper we show that the conjecture is true for several types of maps. In other cases, we provide support for the conjecture by proving that there is a Zariski dense set of points with disjoint orbits and satisfying a_f(P) = d_f.
Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous results, we recover results on arithmetic properties of f-periodic points and non f-periodic points. We also obtain an estimate of growth of heights under f and f^{-1}, which is independently obtained by Lee by a different method.